Thermodynamics of deep geophysical media
V. Pankov, W. Ullmann, R. Heinrich, and D. Kracke

9. Anderson-Grüneisen Parameters

Here, we consider the two useful Anderson-Grüneisen parameters in more detail [Grüneisen, 1926; O. Anderson, 1966a, 1967]: the isothermal dT parameter introduced above (see (31)) and the adiabatic dS parameter defined as


Both parameters are used in geophysical and physical studies [Chung, 1973; Barron, 1979]. The parameter dT is largely applied in analyzing the P-T behavior of a, KT, and t, and dS is used to estimate the temperature dependence of KS and to treat the relations between elastic properties (elastic wave velocities) and thermodynamic data [D. Anderson, 1987; O. Anderson et al., 1987; Isaak et al., 1992; Duffy and Ahrens, 1992a, 1992b; Agnon and Bukowinski, 1990b].

As the temperature decreases in the range T 300 K at P = constant, both parameters dS and dT sharply increase due to decreasing a, but at high temperatures, T > Q, they become more or less constant [O. Anderson et al., 1992a]. From (16), we derive the identity relating dT and dS [Birch, 1952]



which was used to calculate the dT values listed in Table 3 and Pankov et al. [1997]. Deriving (83), we find in passing that





where, for convenience, the notation


is introduced.

The values of L and (partialg/partial T)P at P = 0, calculated by (84) and (85), are also presented in Table 3 and Pankov et al. [1997].

Data for some minerals [O. Anderson et al., 1992a] show that (partialg/partial T)P=0approx 0 over a wide temperature range. Assuming that (partialg/partial T)P = 0 for CV = constant (T > Q), (84) gives


Moreover, since CV = constant, this case results in q = 0, and consequently, according to (35) and (42),


However, the estimation of dT by (42) for qsim 1 is more accurate than the values from (87). Then, one might expect that in (84)


for CV constant.

Using (83), identity (84) can be rearranged to the form



If the last term in (88) is small, then [O. Anderson et al., 1992a]


This approximation is recommended for evaluating of the Anderson-Grüneisen parameters at high temperatures, when there are no sufficient data for applying (83) or (88).

In addition to the analysis of the parameter dT(P, T) described in sections 6 and 8, we consider the following features in the behavior of dT. 1) If g = g(V), then q = q(V), but generally speaking, dT = dT(V, T) since CV = CV(V, T) and Kprime = Kprime(V, T). 2) If CV = CV(T), then (42) holds true, and moreover, g = f(V)/CV(T), q = q(V), although, generally speaking, Kprime = Kprime(V, T) and dT = dT(V, T). Combining the former of these assumptions with the condition KT = KT(V) (i.e., dT = Kprime ), we find


and CV therefore takes the form


In section 6.2.2, the arguments were given for decreasing dT with pressure. The same behavior of this parameter follows from the approximation (42) since both Kprime and q decrease with pressure. The exact relations for the P and T derivatives of dT result from the definition of dT by (31)



If the first term in (91) prevails, we have an unusual case ((partialdT)/(partial P))T > 0. Neglecting the second temperature derivative in (92) (at least at room temperature), we find that (partialdT/partial T)P < 0. However, the approximation dT = Kprime (more realistic at T > Q ) gives, by constast, (partialdT/partial T)P > 0 because of (partial Kprime/partial T)P > 0. Experimental data indicate that ((partial2 KT)/(partial T2))P for T > Q is negative and small in value [O. Anderson et al., 1992a].

Finally, the EOS of type (28), which we used to calculate the thermal expansion coefficient by (36), allows us to determine the explicit pressure (or volume) dependence of KT along an isotherm. Retaining in (36) only the terms involving dKprime0/dT, we obtain



where fprime denote the derivative of f with respect to x. This approximation generalizes the similar Birch formula that follows from (93) for dKprime0/dT = 0 [Birch, 1968]. However, when using usual EOS types, dT from the Birch formula increases with P (except for the Murnaghan EOS for which the behavior of dT depends on the sign of the difference Kprime - dT0 ). Again, we convince ourselves that the term with derivative dKprime0/dT is important in analyzing the thermal expansion by (28).

fig08 Figure 8 shows several curves of dT(x) calculated using (93) and EOS (37), which correspond to the curves of a in Figures 1 and 3 (the straight line dT = 6x - 1 by (47) is also drawn for comparison). The favored value of dKprime0/dT is 2cdot 10-3 K -1 (see section 6.2), and deviations from it substantially affect the dT values at compressions in the lower mantle. At high temperatures, according to (42), we have also the lower limit for dT, dT > Kprime - 1 [O. Anderson et al., 1992a].

10. Bulk Moduli

10.1. P- V- T derivatives

Elastic moduli and their P-V-T derivatives are the characteristics constituting the basis of the Earth's interior thermodynamics [Birch, 1952, 1961; Sumino and O. Anderson, 1984; D. Anderson, 1967, 1987, 1989; O. Anderson et al., 1992a; Duffy and D. Anderson, 1989; Bina and Helffrich, 1992; Duffy and Ahrens, 1992a, 1992b]. These quantities also serve as parameters of EOS's. The simplest estimates of adiabatic and isothermal bulk moduli at high pressure are given by their linear pressure dependences, which, however, begin to overestimate the bulk modulus at a compression of about x < 0.85. The P-T variation of the pressure derivative was considered in many papers devoted to EOS's (see section 3). In order to assess the applicability of empirical EOS's, one often uses a relation between the first Kprimeequiv(partial KT/ partial R)T and second Kprimeprimeequiv (partial2KT/ partial P2)T pressure derivatives at P = 0 [Pankov and Ullmann, 1979a; Jeanloz, 1989; O. Anderson, 1986; Hofmeister, 1991b]. Values of Kprime and KKprimeprime at P = 0 generally lie in the intervals 4-6 and -(5-10), respectively. The uncertainty in (partial KS/partial P)T (measured by ultrasonic or Brillouin scattering methods) can reach 1-5% (with allowance for data from various laboratories) or, in some cases, 20% and even 50%. Anomalous values of Kprime and Kprimeprime are sometimes reported (see references in tables of Pankov et al. [1997]): for example, Kprime = 5-7 (garnet), Kprime = 9-14 (pyroxene), and KKprimeprime = -60 (spinel), which are assumed to take on more usual values as pressure increases.

Let us turn our attention to the relations between the adiabatic-isothermal derivatives of KS and KT. Changing the variables P and S to P and T, we find


where dS is defined by (82). The derivative (partial KS/partial P)S characterizes the curvature of an adiabatic P-V or a Hugoniot curve. Further, from (16)



Eliminating (partialg/partial P)T with the help of (8) and (17) (or (122)) and using (58), we find



Substituting (94) for (partial KS/partial P)T and (93) for dS, we arrive at the Birch [1952] formula



The difference between the adiabatic-isothermal derivatives of KS and KT at normal conditions are generally small (1-2% for mantle minerals). Data and estimations by (94)-(97) show that we usually have


(except for FeO for which (partial KS/partial P)T is poorly known [Pankov et al., 1997]); however, D. Anderson [1989] indicated the inverse inequality


For our high-temperature estimates given in Table 3 and Pankov et al. [1997], it was arbitrarily assumed that


Hence, using (94)-(97), we found substantial differences (up to 10-30%) between the pressure derivatives of KS and KT at high temperatures. In fact, the differences are of the same order of magnitude as the derivative increments due to increasing temperature. Specifically, the estimated partial2KT/partial Ppartial T values are 3.5cdot 10-4 (stishovite), 2cdot 10-4 (ilmenite), 3cdot 10-4 (Mg-perovskite), and 1.7cdot 10-4 (MgO) and do not exceed 1cdot 10-4 for other minerals (although some estimates appear to be negative).

Isaak [1993] estimated partial2KT/partial Ppartial T using an identity of type (91) and Boehler's data on the adiabatic temperature gradient (see also section 12.2). He found partial2 KT/partial Ppartial T = (3.9pm 1.0) cdot 10-4 and (3.3pm 0.9)cdot 10-4 K -1 for MgO and olivine, respectively. Furthermore, he showed this derivative to decrease 30% as the pressure increases isothermally to 100 GPa. A similar order of magnitude was found from shock wave data to be a lower limit for this derivative value [Duffy and Ahrens, 1992a] (see also sections 6 and 9).

In addition to the analysis of the mixed derivatives, we give the following identity



which we derived from (94), using (127) and (partialg/partial T)P from (88). Note that a similar relation of Bukowinski and Wolf [1990] is different from (98) (because of either a reprint or mistake). To give an example, we substitute in (98) the values typical of the lower mantle: (partial KS/partial P)S = 4, agT = 0.1, ((partial lndS)/(partial lnr))S = -1, and dS = 3 for x = 1 and dS 2 for x = 0.7. Then,


which is in agreement with the preceding estimates.

When considering the temperature behavior of KS and KT, the Anderson-Grüneisen parameters dS and dT are represented in the form of (29) [D. Anderson, 1987; Duffy and D. Anderson, 1989; O. Anderson et al., 1992a], which can be rewritten as






In section 9, we considered the principal regularites in the variation of dS and dT with pressure and temperature (some decrease of them with T in the vicinity of T = 300 K, the trend to constant values at T > Q, and the decrease with pressure). Now we dwell on the contributions of the intrinsic dVS and dVT and extrinsic Kprime anharmonic terms in (99) and (100).

D. Anderson [1988, 1989] pointed out that the temperature variation of the bulk modulus at P = constant mostly occurs by the variation in a ; i.e., here, the extrinsic anharmonicity generally prevails and enhances with temperature (due to increasing Kprime ). The estimates given in Table 3 and Pankov et al. [1997] show that, at T = 300 K, we have dVT < 0 (except very uncertain data for FeO); most of the estimates falls into an interval between -1 and -2 (although it was found dVT = -17, -5, and -3.9 for coesite, stishovite, and fictive Fe-perovskite phase, respectively). According to D. Anderson [1989], the values of dVT are typically between -4 and -1 (his table 5, however, contains values outside this interval: 2.2 for orthopyroxene, -5.3 for SrTiO 3, and -19 for CaCO 3 ). The parameter dVS satisfies the inequality |dVS| > 2 for 11 out of 54 minerals considered by D. Anderson (specifically, dVS = -4.1 (GeO 2 ), 3.8 (orthopyroxene), -3.1 (SrTiO 3 ), and -18 (CaCO 3 ). At high temperatures, dVT can be either positive or negative (between -1 and 1), and |dVS| is generally positive, lying in the interval 0-1.5 [Pankov et al., 1997]. Thus, as temperature increases, |dVT|, on average, decreases, but |dVS| increases. The contribution of dVS/dS to (99) is usually less than 10-30% at 300 K and does not exceed 15-60% at high temperatures. Accordingly, dVT contributes no more than 30-40% in (100) at 300 K and usually less than 10% at high temperatures. From this analysis, it follows that the extrinsic anharmonic term, although it generally dominates in the derivatives of KS and KT, decreases its contribution in the case of KS and increases its contribution in the case of KT. The decrease of dVT with temperature leads to the approximation


and additionally, in view of (89) and (99),


Thus, at high temperatures, namely the isothermal, rather than adiabatic bulk modulus becomes depending mostly on volume (i.e., temperature-independent function).

10.2. Interpretation of seismic tomography data

In his analysis of the thermodynamic properties of the lower mantle, D. Anderson [1987, 1988, 1989] relies on seismic tomography and geoid data and assumes that the observed horizontal velocity anomalies are caused by temperature variations. Stacey [1992] showed, however, that it is not possible to explain the anomalies with a purely temperature effect, since in such a case, the geoid highs would be too great. Other hypotheses proposed to interpret the seismic anomalies were related to inhomogeneities of composiion, or partial melting, or even the presence of small amounts of fluids [Price et al., 1989; Duffy and Ahrens, 1992b; and others]. Nevertheless, following D. Anderson, below, we estimate the thermodynamic parameters for the lower mantle, considering the temperature effect formally as a limiting case.

Based on the PREM model, the formula for the acoustic Grüneisen parameter, and tomography data, we have in the lower mantle (partial KS/partial P)S = 3-3.8, g = 1.2pm 0.1, and dS = 1-1.8. Consequently, using (89), (94), (96), (99), and (100), we find dTapprox dS + g = 2.2-3.0, Kprime = 3-3.8 (a small correction can be introduced with the help of a = a0xdT for a0 = 4cdot 10-5 K -1 and Tapprox 2000-3000 K), dVSapprox Kprime - dS = 1.2-2.8, and dVT = dVS- g = -0.1-1.7. If a greater uncertainty is assumed for g, say pm 0.4, then dTsim 1.9-3.3 and dSVsim -0.4-2.0. Thus, although the extrinsic anharmonic effects weaken with pressure, they still prevail under the lower mantle conditions ( Kprime > dSV or |dTV| ). The intrinsic anharmonic term dVS significantly increases with pressure, but its isothermal analog dVT can either increase or decrease and reach zero. D. Anderson, by reference to experimental data, points out the case of dVTapprox 0, which yields dVSsim g = 1.2.

However, this value of dSV disagrees with data for such a representative lower-mantle material as periclase. Using dVT = 0 and dT = Kprimeapprox 3.2-3.5 for x = 0.7 and the Birch-Murnaghan EOS for MgO, we find dSsim dT- g = 2.2-2.5 for g = g0xq = 1, see (119)). These values of dT and dS are, on average, still exceed the results inferred from seismic models. O. Anderson et al. [1992b] noted that data for MgO can be reconciled with seismic results by assuming that q < 1. In particular, our analysis leads to the following consistent sequence of values: dVT 0.2, q 0.8, g = g0xq 1.1 (for x = 0.7 ), dT = Kprime- dVT 3-3.3, and dS 1.9-2.2. In any case, the consistency to seismic data could be found in this manner if the values derived from seismic tomography were explained by only horizontal variations of temperature.

10.3. Estimation of KS at high temperature


The consideration of the temperature behavior of KS at P = 0 will be added by the following two methods. One of them uses the condition dS = dastS = constant [O. Anderson, 1988; Duffy and D. Anderson, 1989], and in view of (99), yields the power law


which we used to estimate the values listed in Table 2 and Pankov et al. [1997].

Another approach proposed by O. Anderson [1989] and extended by O. Anderson et al. [1992a] is based on data for enthalpy. We obtain the relation of KS to enthalpy using a somewhat different procedure, namely, the formula g = aKSV/CP from which the derivative (partial KS/partial H)P is found, and thus, approximately,


where the asterisk marks the values at a fixed temperature. By using the parameter values from Table 2 and Pankov et al. [1997], as well as data for enthalpy, we estimated KS by (104) for a number of minerals (Table 4). One can see that the O. Anderson's method is quite efficient: the uncertainty of the estimated values at high temperatures is 2-5%. It is also clear that both methods described above would give more accurate results when high-temperature values for KSast, rast, gast, Hast, and dSast are used in the respective formulas.


fig09 In conclusion to this analysis, we show the dependence of KS versus T at P = 0 (Figure 9) calculated by the Mie-Grüneisen EOS with g = g0xq for three minerals considered in sections 6-8. Comparing the KS curves for various values of q = 0-2 with experimental data, we see that it is possible to choose appropriate values of q consistent to the data. However, considering the results presented for the same EOS's in sections 6-8, it is not always possible to find such values of q for which the EOS becomes consistent to data simultaneously for a, CP, t, and KS. Thus, as in sections 6-8, we conclude that the thermal part of the Mie-Grüneisen EOS does not provide suffuciently high accuracy of all the thermodynamic parameters calculated from this EOS.

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